Heisenberg equation

Using the Heisenberg equation of motion, (AS,2,40). the connnutator in the last expression may be replaced by the time-derivative operator... 

B) THE MICROSCOPIC HYPERPOLARIZABILITY IN TERMS OF THE LINEAR POLARIZABILITY THE KRAMERS-HEISENBERG EQUATION AND PLACZEK LINEAR POLARIZABILITY THEORY OF THE RAMAN EFFECT... 

As long as the system can be described by the rate constant - this rules out the localization as well as the coherent tunneling case - it can with a reasonable accuracy be considered in the imaginary-time framework. For this reason we rely on the Im F approach in the main part of this section. In a separate subsection the TLS real-time dynamics is analyzed, however on a simpler but less rigorous basis of the Heisenberg equations of motion. A systematic and exhaustive discussion of this problem may be found in the review [Leggett et al. 1987]. 

Solving now the Heisenberg equations of motion for the a operators perturbatively in the same way as in the weak-coupling case, one arrives (at = 0) at the celebrated non-interacting blip approximation [Dekker 1987b Aslangul et al. 1985]... 

The u and v representations are sometimes distinguished as the Schrodinger and the Heisenberg representation. For stationary operators P, then, the Heisenberg equation of motion is... 

We shall again postulate commutation rules which have the property that the equations of motion of the matter field and of the electromagnetic field are consequences of the Heisenberg equation of motion ... 

Since the original operator Q(t) obeyed the Heisenberg equation of motion... 

Thus Ql —0 does satisfy the correct Heisenberg equation of motion. It should be recalled, incidentally, that the correct definition of the adjoint A of an antilinear operator A is... 

Heisenberg equation of motion for stationary operators, 413 Heisenberg field, transformation properties, 691... 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

In this volume dedicated to Yngve Ohm we feel it is particularly appropriate to extend his ideas and merge them with the powerful practical and conceptual tools of Density Functional Theory (6). We extend the formalism used in the TDVP to mixed states and consider the states to be labeled by the densities of electronic space and spin coordinates. (In the treatment presented here we do not explicitly consider the nuclei but consider them to be fixed. Elsewhere we shall show that it is indeed straightforward to extend our treatment in the same way as Ohm et al. and obtain equations that avoid the Bom-Oppenheimer Approximation.) In this article we obtain a formulation of exact equations for the evolution of electronic space-spin densities, which are equivalent to the Heisenberg equation of motion for the electtons in the system. Using the observation that densities can be expressed as quadratic expansions of functions, we also obtain exact equations for Aese one-particle functions. 

The time-dependent Eq. (4.5) in terms of the density is exact and equivalent to the full Heisenberg equation of motion when no approximations or models are invoked. It is thus worthwhile to display it in more detail... 

The Heisenberg equation for the population operator is coupled to analogous equations for the free-bound coherences with n) = 1), 2) ... 

The equations of motion for NGF are obtained from the Heisenberg equation of motion for operators... 

Following the same way, as for the retarded functions (using only the definitions of NGF and Heisenberg equations of motion) one derives instead of (384)-(386)... 

In the following, we indicate the time derivative of a hermitian operator B with the symbol B. In the Heisenberg representation of quantum mechanics, it obeys the Heisenberg equation of motion... 

Dirac s development of TDHF theory invokes the Heisenberg equation of motion for the density operator as a basic postulate,... 

The ACF of the dipole moment operator of the fast mode may be written in the presence of Fermi resonances by aid of Eq. (10). Besides, the dipole moment operator at time t appearing in this equation is given by a Heisenberg equation involving the full Hamiltonian (225). The thermal average involved in the ACF must be performed on the Boltzmann operator of the system involving the real... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

In Eq. (Ill), the average is taken over QD and SM electronic and vibrational states. Then, making use of the Heisenberg equation, we obtain the following expression for the photocurrent [45,52] ... 

Using (4), the Heisenberg equation for creation and annihilation operators can be presented in the following form ... 

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